Universitat des Saarlandes

UN

IVE R S IT

AS

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RA V I E N

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Fachrichtung 6.1 – Mathematik

Preprint Nr. 309

Subsonic phase transition wavesin bistable lattice models with small spinodal region

Michael Herrmann, Karsten Matthies,Hartmut Schwetlick and Johannes Zimmer

Saarbrucken 2012

Fachrichtung 6.1 – Mathematik Preprint No. 309Universitat des Saarlandes submitted: June 1, 2012

Subsonic phase transition wavesin bistable lattice models with small spinodal region

Michael HerrmannUniversitat des Saarlandes, Fachrichtung Mathematik

michael.herrmann@math.uni-sb.de

Karsten MatthiesUniversity of Bath, Department of Mathematical Sciences

k.matthies@maths.bath.ac.uk

Hartmut SchwetlickUniversity of Bath, Department of Mathematical Sciences

h.schwetlick@maths.bath.ac.uk

Johannes ZimmerUniversity of Bath, Department of Mathematical Sciences

zimmer@maths.bath.ac.uk

Edited byFR 6.1 – MathematikUniversitat des SaarlandesPostfach 15 11 5066041 SaarbruckenGermany

Fax: + 49 681 302 4443e-Mail: preprint@math.uni-sb.deWWW: http://www.math.uni-sb.de/

Subsonic phase transition waves

in bistable lattice models with small spinodal region

Michael Herrmann∗, Karsten Matthies†, Hartmut Schwetlick‡, Johannes Zimmer§

June 1, 2012

Abstract

Phase transitions waves in atomic chains with double-well potential play a fundamental role inmaterials science, but very little is known about their mathematical properties. In particular, theonly available results about waves with large amplitudes concern chains with piecewise-quadraticpair potential. In this paper we consider perturbations of a bi-quadratic potential and prove thatthe corresponding three-parameter family of waves persists as long as the perturbation is smalland localised with respect to the strain variable. More precisely, we introduce an anchor-correctoransatz, characterise the corrector as a fixed point of a nonlinear and nonlocal operator, and showthat this operator is contractive in a small ball of a certain function space.

Keywords: phase transitions in lattices, heteroclinic travelling waves,FPU-type chains, kinetic relations

MSC (2010): 37K60, 47H10, 74J30, 82C26

Contents1 Introduction 1

1.1 Overview and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Preliminaries and reformulation of the problem 52.1 Reformulation as integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Transformation to the special case Iδ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Proof of the existence and uniqueness result 63.1 Inversion formula for M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Solution operator to the linear subproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Properties of the nonlinear operator G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Fixed point argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Change in the kinetic relation 13

1 Introduction

Many standard models in one-dimensional discrete elasticity describe the motion in atomic chainswith nearest neighbour interactions. The corresponding equation of motion reads

uj(t) = Φ′(uj+1(t)− uj(t))− Φ′(uj(t)− uj−1(t)) , (1)

where Φ is the interaction potential and uj denotes the displacement of particle j at time t.Of particular importance is the case of non-convex Φ, because then (1) provides a simple dy-

namical model for martensitic phase transitions. In this context, a propagating interface can be

∗michael.herrmann@math.uni-sb.de, Universitat des Saarlandes†k.matthies@maths.bath.ac.uk, University of Bath‡h.schwetlick@maths.bath.ac.uk, University of Bath§zimmer@maths.bath.ac.uk, University of Bath

1

described by a phase transition waves, which is a travelling wave that moves with subsonic speed andis heteroclinic as it connects periodic oscillations in different wells of Φ. The interest in such wavesis also motivated by the quest to derive selection criteria for the naıve continuum limit of (1), whichis the PDE ∂ttu = ∂xΦ′(∂x). This equation is ill-posed for nonconvex Φ due to its elliptic-hyperbolicnature, and one proposal is to select solutions by so-called kinetic relations [AK91, Tru87] derivedfrom travelling waves in atomistic models.

Combining the travelling wave ansatz uj(t) = U(j − ct) with (1) yields the delay-advance-differential equation

c2R′′(x) = ∆1Φ′(R(x)), (2)

where R(x) := U(x + 1/2) − U(x − 1/2) the (symmetrised) discrete strain profile and ∆1F (x) :=F (x+ 1)−2F (x) +F (x−1). Periodic and homoclinic travelling waves have been studied intensively,see [FW94, SW97, FP99, Pan05, Her10] and the references therein, but very little is known aboutheteroclinic waves. The authors are only aware of [HR10, Her11], which prove the existence ofsupersonic heteroclinic waves, and the small amplitude results from [Ioo00]. In particular, thereseems to be no result that provides phase transitions waves with large amplitudes for generic double-well potentials.

Phase transition waves with large amplitudes are only well understood for piecewise quadraticpotentials, and there exists a rich body of literature on bi-quadratic potentials, starting with [BCS01a,BCS01b], or tri-quadratic potentials [Vai10]. For the special case

Φ(r) = 12r

2 − |r| , Φ′(r) = r − sgn(r) (3)

the existence of phase transition waves has been established by two of the authors using rigorousFourier methods. In [SZ09] they consider subsonic speeds c sufficiently close to 1, which is the speedof sound, and show that (2) admits for each c a two-parameter family of waves. These waves haveexactly one interface and connect different periodic tail oscillations, see Figure 2 for an illustration.

In this paper we allow for small perturbations of the potential (3) and show that the phasetransition waves from [SZ09] persist provided that the perturbation is sufficiently small and localised.Our approach is in essence perturbative and reformulates the travelling wave equation with perturbedpotential in terms of a corrector profile S. The resulting equation can be written as

MS = A2G(S) + η , (4)

where η is a constant of integration. Moreover, M and A are linear integral operators and G anonlinear superposition operator to be identified below. The analysis of (4) is rather delicate sincethe Fourier symbol of M has real roots. In our existence proof, we first eliminate the singularitiesand derive an appropriate inversion formula for M. Afterwards we introduce a class of admissiblefunctions S and show that the properties of A and G guarantee that A2G(S) is compactly supportedand sufficently small. These fine properties are illustrated in Fig. 4 and allow us to define a nonlocaland nonlinear operator T such that

MT (S) = A2G(S) + η(S)

holds for all admissible S with some η(S) ∈ R. In the final step, we show that T is contractive insome ball of an appropriately defined function space.

We further remark that each phase transition wave satisfies Rankine-Hugoniot conditions forthe macroscopic averages of mass, momentum, and total energy [HSZ12], which imply nontrivialrestrictions between the wave speed and the tail oscillations on both sides of the interface. Althoughthese conditions do not appear explicitly in our existence proof, they can (at least in principle) becomputed because the tail oscillations are given by harmonic waves, see again Fig. 2. For generaldouble-well potentials, however, it is much harder to evaluate the Rankine-Hugoniot conditions andthus it remains unclear which tail oscillations can be connected by phase transition waves. Closelyrelated to the jump condition for the total energy is the kinetic relation, which specifies the transferbetween oscillatory and non-oscillatory energy at the interface and determines the configurational

2

force that drives the wave. In the final section we discuss how the kinetic relation changes to leadingorder under small perturbation of the potential (3).

We now present our main result in greater detail.

1.1 Overview and main result

We study an atomic chain with interaction potential

Φδ(r) = 12r

2 −Ψδ(r) , Ψδ(0) = 0 ,

where Ψ′δ is a perturbation of Ψ′0 = sgn in a small neighbourhood of 0. The travelling wave equationtherefore reads

c2R′′ = 41

(R−Ψ′δ(R)

)(5)

and depends on the parameters c and δ. In order to show that (5) admits solutions for small δ werely on the following assumptions on Ψ′δ, see Figure 1 for an illustration.

Assumption 1. Let (Ψδ)δ>0 be a one-parameter family of C2-potentials such that

(i) Ψ′δ coincides with Ψ′0 outside the interval (−δ, δ),

(ii) there is a constant CΨ independent of δ such that∣∣Ψ′δ(r)∣∣ ≤ CΨ ,∣∣Ψ′′δ (r)∣∣ ≤ CΨ

δ

for all r ∈ R.

The quantity

Iδ := 12

ˆR

(Ψ′δ(r)−Ψ′0(r)

)dr

plays in important role in our perturbation result as it determines the leading order correction.Notice that our assumptions imply

Iδ = 12

ˆ δ

−δΨ′δ(r) dr = −1

2(Φδ(+1)− Φδ(−1)) and hence |Iδ| ≤ CΨδ .

As already mentioned, the case δ = 0 has been solved in [SZ09]. The main result can be summarisedas follows.

Proposition 2 ([SZ09], Proof of Theorem 3.11). There exist 0 < c0 < 1 such that for every c ∈ [c0, 1),there exists a two-parameter family of solutions R0 ∈ W2,∞(R) to the travelling wave equation (5)with δ = 0 such that

(i) R0(0) = 0 ,

(ii) ‖R0‖∞ ≤ D0

(1− c2

)−1,

(iii) R0(x) > r0 for x > x0 and R0(x) < −r0 for x < −x0 ,

(iv) R′0(x) > d0 for |x| < x0 ,

for some constants x0, r0, d0 and D0 which depend only on c0. For any such profile R0 there existsconstants α± and β± such that

limx→±∞

∣∣±rc + α±(1− cos (kcx)) + β± sin (kcx)−R0(x)∣∣ = 0 ,

where rc and kc > 0 are uniquely determined by c. Moreover, any two profiles R0 and R0 satisfy

R0 − R0 ∈ span {sin (kc·), 1− cos (kc·)} .

3

r

rr

0�(r)

+� +�

��(r)

����

+ 12

� 12

+1

�1

Figure 1: Sketch of Ψ′δ and Φδ for δ = 0 (grey) and δ > 0 (black). Since Φ0 is symmetric, −Iδ is justhalf the energy difference between the two wells of Φδ.

graph of R

Figure 2: Sketch of the waves for δ = 0 (grey) and δ > 0 (black) as provided by our perturbationresult; the shaded region indicates the spinodal interval [−δ, +δ], where Ψ′δ differs from Ψ′0. Bothwaves differ by the constant Iδ = O(δ) and a small oscillatory corrector S of order O

(δ2). The tail

oscillations of both waves do not penetrate the spinodal region and are generated by harmonic waveswith wave number kc. Notice that the phase shifts, amplitudes, and local averages are different onboth sides of the interface and depend on δ.

The main result of this article can be described as follows.

Theorem 3. For all c1 ∈ (c0, 1) there exists δ0 > 0 such that for any 0 < δ < δ0, any speedc0 < c < c1, and any given wave R0 as in Proposition 2 there exists a solution R to (5) with

R = R0 − Iδ + S,

where Iδ = O(δ) and the corrector S ∈W2,∞(R)

(i) vanishes at x = 0,

(ii) is small in the sense of

‖S‖∞ = O(δ2), ‖S′‖∞ = O(δ), ‖S′′‖∞ = O(1).

Moreover, for small δ there exists only one R with these properties.

Concerning these assertions, we emphasise that:

(i) Our existence proof implies that different choices of R0 provide different waves R, see Lemma 15.

(ii) All constants derived below depend on c1 and c0 but for notational simplicity we do not writethis dependence explicitly. It remains open whether δ0 can be chosen independently of c1.

(iii) The surprisingly simple leading order effect, that is the addition of −Iδ to R0, implies thatthe kinetic relation does not change to order O(δ). Notice, however, that the kinetic relationdepends on the choice of R0, cf. [SZ].

(iv) The travelling wave equation (5) is, of course, invariant under shifts in x but fixing R0 and Sat 0 removes neutral directions in the contraction proof.

4

This paper is organised as as follows. In §2 we reformulate (5) in terms of integral operators A andM and show that it is sufficient to prove the existence of waves for the special case Iδ = 0. In Section3 we then study the equation for the corrector S. We first establish an inversion formula forM andinvestigate afterwards the properties of the nonlinear G. These result then allow us to establish thefixed point argument for the operator T . Finally, we discuss the kinetic relation in §4.

2 Preliminaries and reformulation of the problem

In this section we reformulate the travelling wave equation (5) in terms of integral operators andshow that elementary transformations allow us to assume that Iδ = 0 holds for all δ > 0.

2.1 Reformulation as integral equation

For our analysis it is convenient to reformulate the problem in terms of the convolution operator Aand the operator M defined by

(AF )(x) :=

x+1/2ˆ

x−1/2

F (s) ds , MF := A2F − c2F .

The travelling wave equation can then be stated as

MR = A2Ψ′δ(R) + µ . (6)

Lemma 4. A function W ∈ W2,∞(R) solves the travelling wave equation (5) if and only if thereexists a constant µ ∈ R such that (R, µ) solves (6).

Proof. By definition of A, we have d2

dx2A2 = 41. Equation (5) is therefore, and due to the definition

of M, equivalent to

(MR)′′ = P ′′ , P := A2Ψ′δ(R) . (7)

The implication (6) =⇒ (5) now follows immediately. Towards the reversed statement, we integrate(7)1 twice with respect to x and obtain MR = P + λx + µ, where λ and µ denote constants ofintegration. The condition R ∈ L∞(R) implies MR, Ψ′δ(R), P ∈ L∞(R), and we conclude thatλ = 0.

We next summarize some properties of the operator A.

Lemma 5. For any 1 ≤ p ≤ ∞ we have A : Lp(R)→W1,p(R) ∩ BC(R) with

‖AF‖Lp(R) ≤ ‖F‖Lp(R), ‖(AF )′‖Lp(R) ≤ 2‖F‖Lp(R), ‖AF‖BC(R) ≤ ‖F‖Lp(R)

for all F ∈ Lp(R), where (AF )′ = ∇F = F(·+ 1

2

)− F

(· − 1

2

). Moreover, suppF ⊂ [x1, x2] implies

suppAF ⊂ [x1 − 12 , x2 + 1

2 ].

Proof. All assertions follow immediately from the definition of A and Holder’s inequality.

Some of the arguments in our existence proof rely on Fourier transform (in the space of tempereddistributions), which we normalise as follows

F (k) =1√2π

ˆReikxF (x) dx , F (x) =

1√2π

ˆRe−ikxF (k) dk .

In particular, the operators A and M diagonalise in Fourier space and have symbols

a(k) =sin (k/2)

k/2and m(k) = a(k)2 − c2 ,

respectively.

5

2.2 Transformation to the special case Iδ = 0

The key observation that traces the general case Iδ 6= 0 back to the special case Iδ = 0 is that anyshift in Ψ′δ can be compensated by adding a constant to R.

Lemma 6. The family (Ψδ)δ>0 defined by

δ = δ(1 + CΨ), Ψ′δ(r) = Ψ′δ(r − Iδ)

satisfies Assumption 1 with constant CΨ = CΨ(1 + CΨ) as well as

Iδ = 12

ˆR

Ψ′δ(r)−Ψ′0(r) dr = 0 for all δ > 0.

Moreover, each solution (R, µ) to the modified travelling wave equation

MR = A2Ψ′δ(R) + µ (8)

defines a solution (R, µ) to (6) via R = R− Iδ and µ = µ−(c2 − 1

)Iδ and vice versa.

Proof. Due to |Iδ| ≤ CΨδ and our definitions we find Ψ′δ(r) = Ψ′0(r) at least for all r with |r| ≥ δ, as

well as ∣∣∣Ψ′δ(r)∣∣∣ ≤ CΨ ≤ CΨ ,

∣∣∣Ψ′′δ(r)∣∣∣ ≤ CΨ

δ=CΨ

δ

1 + CΨ

1 + CΨ=CΨ

δfor all r ∈ R .

We also have

Iδ = 12

ˆR

(Ψ′δ(r − Iδ)−Ψ′0(r)

)dr = 1

2

ˆR

(Ψ′δ(r)−Ψ′0(r + Iδ)

)dr

= 12

ˆR

(Ψ′δ(r)−Ψ′0(r)

)dr + 1

2

ˆR

(Ψ′0(r)−Ψ′0(r + Iδ)

)dr = Iδ − Iδ = 0 .

Finally, the equivalence of (6) and (8) is obvious.

3 Proof of the existence and uniqueness result

We now show that for fixed c the two-parameter family of phase transitions for δ = 0 persists underthe perturbation Ψ0 Ψδ provided that δ is sufficiently small. To this end we proceed as follows.

(i) We fix c with c0 < c < c1 < 1 with c0 and c1 as in Proposition 2 and Theorem 3. Then thereexists a unique solution kc > 0 to a(kc) = c, and this implies m(±kc) = 0, m′(±kc) 6= 0, andm(k) 6= 0 for k 6= ±kc. We emphasise again that all constants derived below can be chosenindependently of c but are allowed to depend on c0 and c1.

(ii) Thanks to Proposition 2 and Lemma 4, we fix (R0, µ0) from the two-parameter family ofsolutions to the integrated travelling wave equation (6) for δ = 0 and given c. Recall that R0

is normalised by R0(0) = 0.

(iii) In view of Lemma 6, we assume that Iδ = 0 holds for all δ > 0. To avoid unnecessarytechnicalities we also assume from now on that δ is sufficiently small.

In order to find a solution (R, µ) to the integrated travelling wave equation (6) for δ > 0, we furthermake the anchor-corrector ansatz

R = R0 + S , µ = µ0 + η ,

6

and seek correctors (S, η) such that

MS = A2G+ η , G = G(S) . (9)

Here, the nonlinear operator G is defined by

G(S)(x) = Ψ′δ(R0(x) + S(x)

)−Ψ′0

(R0(x)

).

A natural ansatz space for S is given by

X :={S ∈W2,∞(R) : S(0) = 0

},

where we impose the constraint S(0) = 0 in order to eliminate the non-uniqueness resulting from theshift invariance of the travelling wave equation (6). In fact, without this normalisation condition anycorrector S provides a whole family of other possible correctors via S = S(·+ x0) +R0(·+ x0)−R0

with x0 = O(δ2).

3.1 Inversion formula for M

Our first task is to construct for given G a solution (S, η) to the affine equation (9)1. In a preparatorystep, we therefore derive an appropriate inversion formula for the operator M. Recall that theredoes not exist a unique inverse of M as the corresponding Fourier symbol, which is the function m,has real roots at k = ±kc.

graph of Yi graph of MYi graph of

bYi

+kc5.0

5.0 �kc1.0

Figure 3: Properties of Y1 (grey) and Y2 (black).

As illustrated in Figure 3, we introduce two functions Y1, Y2 ∈ L∞(R) by

Y1(x) :=

√2π

m′(kc)cos (kcx)sgn(x) , Y2(x) :=

√2π

m′(kc)sin (kcx)sgn(x) ,

and verify by direct computations the following assertions.

Remark 7. We have

(i) MYi ∈ L∞(R) with suppMYi ⊆ [−1, 1] ,

(ii) Y1(k) = +2i

m′(kc)

k

k2 − k2c

and Y2(k) = − 2

m′(kc)

kck2 − k2

c

,

(iii) mYi ∈ L2(R) ∩ BC1(R) with limk→±kc

m(k)Y1(k) = ±i and limk→±kc

m(k)Y2(k) = −1.

Using Y1 and Y2, we can establish the following linear and continuous inversion formula for M.

Lemma 8. Let Q be given with Q ∈ L2(R) ∩ BC1(R). Then there exists Z ∈ L2(R), which dependslinearly on Q, such that

M

(Z − i

Q(+kc)− Q(−kc)2

Y1 −Q(+kc) + Q(−kc)

2Y2

)= Q , (10)

7

and

‖Z‖2 ≤ D(‖Q‖2 + ‖Q‖1,∞

)for some constant D independent of Q.

Proof. Thanks to our assumptions on Q and the properties of m and Y1,2, the function Z ∈ L2(R) iswell defined by

Z(k) :=Q(k) + i

Q(+kc)− Q(−kc)2

m(k)Y1(k) +Q(+kc) + Q(−kc)

2m(k)Y2(k)

m(k)∈ L2(R) ∩ BC(R)

and satisfies (10) by construction. Moreover, with J := [−2kc, +2kc] we readily verify the estimates

‖Z‖L2(R\J) ≤ ‖m−1‖L∞(R\J)‖Q‖L2(R\J) +(∣∣Q(+kc)

∣∣+∣∣Q(−kc)

∣∣)(‖Y1‖L2(R\J) + ‖Y2‖L2(R\J)

)≤ D

(‖Q‖L2(R) + ‖Q‖BC(R)

)and ‖Z‖C(J) ≤ D‖Q‖C1(J).

We note that the constant D in Lemma 8 is uniform in c0 < c < c1 but will grow with c1 → 1,due to the definition of Y1 and Y2 and the properties of m.

3.2 Solution operator to the linear subproblem

We are now able to prove that the affine problem (9)1 has a nice solution operator on the space ofall compactly supported functions G.

Lemma 9. Let G ∈ L∞(R) be given, with supp G ⊆ [−1, 1]. Then there exist S ∈ X and η ∈ R, bothdepending linearly on G, such that

MS = A2G+ η .

Moreover, we have

(i) |η| ≤ CM‖A2G‖∞ ,

(ii) ‖S‖∞ ≤ CM‖A2G‖∞ ,

(iii) ‖S′‖∞ ≤ CM‖AG‖∞ ,

(iv) ‖S′′‖∞ ≤ CM‖G‖∞.

for some constant CM > 0 independent of G.

Proof. The function Q := A2G satisfies supp Q ⊆ [−2, 2], and using

∣∣∣Q(k)∣∣∣+

∣∣∣∣ ddk Q(k)

∣∣∣∣ ≤ C2ˆ

−2

(1 + |x|

)|Q(x)| dx ≤ C‖Q‖L∞(R)

as well as ‖Q‖2 = ‖Q‖2, we easily verify that

‖Q‖2 + ‖Q‖1,∞ ≤ C‖Q‖∞ .

Using Lemma 8, we now define S := Z + f1Y1 + f2Y2 such that MS = Q, where Z ∈ L2(R), and

‖Z‖2 + |f1|+ |f2| ≤ C‖Q‖∞ = C‖A2G‖∞ . (11)

8

By definition of M, Q, and S we have

c2S = −A2G+A2S = −A2G+A2(Z + f1Y1 + f2Y2

), (12)

and the properties of A imply

‖A2Z‖∞ ≤ ‖Z‖2, ‖A2Yi‖∞ ≤ ‖Yi‖∞ .

Combining these estimates with (11) and (12), we arrive at S ∈ L∞(R) with

‖S‖∞ ≤ C‖A2G‖∞ .

Moreover, differentiating the first identity in (11) with respect to x, we get

c2S′ = ∇(−AG+AS

), c2S′′ = ∇∇

(−G+ S

),

where the discrete differential operator ∇ is defined as ∇U = U(·+ 1

2

)− U

(· − 1

2

), cf. Lemma 5.

This implies

‖S ′‖∞ ≤ C‖AG‖∞ , ‖S ′′‖∞ ≤ C‖G‖∞

thanks to ‖A2G‖∞ ≤ ‖AG‖∞ ≤ ‖G‖∞ and ‖AS‖∞ ≤ ‖S‖∞. We finally define S(x) = S(x)− S(0)and η =

(1− c2

)S(0). All assertions now follow by taking CM as the maximum of all constants C

in this proof.

While we will work in a subset of W2,∞, we remark that the regularity of the equation(Lemma 9 (iv)) would allow us to start the iteration with S ∈W1,∞.

3.3 Properties of the nonlinear operator G

We say that S ∈ X is δ-admissible if there exist two number x− < 0 < x+, which both depend on Sand δ, such that

(i) R0(x±) + S(x±) = ±δ ,

(ii) R0(x) + S(x) < −δ for x < x− ,

(iii) R0(x) + S(x) > +δ for x > x+ ,

(iv) 12R′0(0) < R′0(x) + S′(x) < 2R′0(0) for x− < x < x+ ,

Notice that the properties of R0 implies that the set of admissible correctors is not empty for allsufficiently small δ.

We are now able to derive the second key argument for our fixed-point argument.

Lemma 10. Let S be δ-admissible and G = G(S). Then we have

‖G‖∞ ≤ D, supp G ⊆ [−Dδ, Dδ],ˆRG(x) dx ≤ D

(1 + ‖S′′‖∞

)δ2

for some constant D independent of S and δ.

Proof. The first assertion is a consequence of ‖G‖∞ ≤ 1 + CΨ. Since S is δ-admissible, we have

suppG = [x−, x+] , ±δ = ±ˆ x±

0

(R′0(x) + S′(x)

)dx

9

with x± as above, and this implies

1

2R′0(0)δ ≤ |x±| ≤

2

R′0(0)δ, suppG ⊆ 2

R′0(0)[−δ, δ] . (13)

Using the Taylor estimate∣∣R′0(x) + S′(x)−R′0(0)− S′(0)∣∣ ≤ (‖R′′0‖∞ + ‖S′′‖∞

)|x| , (14)

we also verify that∣∣∣∣x± ∓ δ

R′0(0) + S′(0)

∣∣∣∣ ≤ |x±|22

‖R′′0‖∞ + ‖S′′‖∞R′0(0) + S′(0)

≤ 4δ2 ‖R′′0‖∞ + ‖S′′‖∞R′0(0)3 . (15)

A direct computation now yields

ˆRG(x) dx =

ˆ x+

x−

Ψ′δ(R0(x) + S(x)

)dx−

ˆ x+

x−

sgn(x) dx =

ˆ δ

−δΨ′δ(r)

dr

z(r)− |x+ + x−| , (16)

where z(R0(x) + S(x)) = R′0(x) + S′(x) for all x ∈ [x−, x+]. Thanks to (14), our assumption

Iδ =´ +δ−δ Ψ′δ(r) dr = 0, and the estimate z(r), z(0) ≥ 1

2R′0(0) we get

∣∣∣∣ˆ δ

−δΨ′δ(r)

dr

z(r)

∣∣∣∣ ≤ ˆ δ

−δ

∣∣Ψ′δ(r)∣∣ |z(r)− z(0)|z(r)z(0)

dr ≤ Dδ(|x+|+ |x−|

)(‖R′′0‖∞ + ‖S′′‖∞

)R′0(0)2 .

and combining this with (13), (15) and (16) gives∣∣∣∣ˆRG(x) dx

∣∣∣∣ ≤ Dδ2 ‖R′′0‖∞ + ‖S′′‖∞R′0(0)3 . (17)

By Proposition 2 (iv), R′0(0) is bounded from below. Moreover, combining Proposition 2 (ii) withthe equation for R′′0 , that is

c2R′′0 = ∆1R0 −∆1sgn ,

we find a constant D, which depends only on c0 and c1, such that ‖R′′0‖∞ ≤ D. The second andthird assertion are now direct consequences of these observations and the estimates (15) and (17).

Corollary 11. There exists a constant CG, which is independent of δ, such that

‖AG‖∞ ≤ CGδ , ‖A2G‖∞ ≤ CG(1 + ‖S′′‖∞

)δ2 ,

holds with G = G(S) for all δ-admissible S.

Proof. Thanks to Lemma 10 and the properties of A, we find some constant D such that

|AG(x)| ≤ Dδ for |x± 12 | ≤ Dδ

AG(x) =´RG(x) dx for |x| ≤ 1

2 −Dδ ,AG(x) = 0 for |x| ≥ 1

2 +Dδ ,

see Figure 4 for an illustration. The first estimate is now a consequence of the trivial estimate∣∣´RG(x) dx

∣∣ ≤ |supp G| ‖G‖∞ ≤ Dδ, whereas the second one follows from

∣∣(A2G)(x)∣∣ ≤ Dδ2 +

∣∣∣∣ˆRG(x) dx

∣∣∣∣ for all x ∈ R

and the refined estimate∣∣´

RG(x) dx∣∣ ≤ D(1 + ‖S‖′′∞)δ2.

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0.5 1.0

O(1) O(�) O(�2)

graph of Ggraph of AG graph of A2G

1.0O(�2)

1.0

Figure 4: Properties of G = G(S) for δ–admissible S. The shaded regions indicate intervals withlength of order O(δ).

In the general case Iδ 6= 0, one finds – due to´RG(x) dx = 2Iδ + O(δ2) – the weaker estimate

‖A2G‖∞ ≤ D(1 + ‖S′′∞‖)δ. This bound is still sufficient to establish a fixed point argument, butprovides only a corrector S of order O(δ). Recall, however, that Lemma 6 shows that shifting Ψδ

and changing R0 allows us to find correctors of order O(δ2) even in the case Iδ 6= 0.

We finally derive continuity estimates for G.

Lemma 12. There exists a constant CL independent of δ such that

‖A2G1 −A2G2‖∞ + ‖AG1 −AG2‖∞ + δ‖G1 −G2‖∞ ≤ CL(δ‖S′1 − S′2‖∞ + δ2‖S′′1 − S′′2‖∞

)holds for all δ-admissible correctors S1 and S2 with G` = G(S`).

Proof. According to Lemma 10, there exists a constant D, such that G`(x) = 0 for all x with|x| ≥ Dδ. For |x| ≤ Dδ, we use Taylor expansions for S1 − S2 at x = 0 to find

|S2(x)− S1(x)| ≤ ‖S′2 − S′1‖∞ |x|+ ‖S′′2 − S′′1‖∞ |x|2 ,

where we used that S2(0)−S1(0) = 0. Combining this estimate with the upper bounds for |Ψ′′δ | gives∣∣G1(x)−G2(x)∣∣ ≤ D

δ

∣∣S1(x)− S2(x)∣∣ ≤ D(‖S′1 − S′2‖∞ + δ‖S′′1 − S′′2‖∞

),

and hence the desired estimate for ‖G2 −G1‖∞. We also have

‖A2G1 −A2G2‖∞ ≤ ‖AG1 −AG2‖∞ ≤ |supp(G2 −G1)| ‖G2 −G1‖∞ ≤ Dδ‖G2 −G1‖∞ ,

and this completes the proof.

3.4 Fixed point argument

Now we have prepared all ingredients to establish a suitable fixed point argument in the space

Xδ :={S ∈ X : ‖S‖∞ ≤ C0δ

2, ‖S′‖∞ ≤ C1δ, ‖S′′‖∞ ≤ C2

}with constants

C2 := CM(1 + CΨ), C1 := CMCG , C0 := CMCG(1 + C2) .

Lemma 13. For all sufficiently small δ there exists an operator

T : Xδ → Xδ

such that for any S ∈ Xδ we have

MT (S) = A2G(S) + η(S)

for some η(S) ∈ R with |η(S)| ≤ C0δ2.

11

Proof. Step 1: We first show that each S ∈ Xδ is δ-admissible provided that δ is sufficiently small.According to Proposition 2, there exist positive constants r0, x0, and d0 such that∣∣R0(x)

∣∣ ≥ r0 for |x| > x0 , d0 < R′0(x) for |x| < x0 ,

and combining the upper estimate for ‖R0‖∞ with the equation for R0 we find ‖R′′0‖∞ ≤ D2 for someconstant D2. We now set

δ0 :=1

2min

{d0

2D2d0

+ C1

, x0d0,

√r0

C0, r0

}, xδ :=

2

d0δ

and assume that δ < δ0. For any x with |x| ≤ xδ ≤ x0, we then estimate∣∣R′0(x) + S′(x)−R′0(0)∣∣ ≤ D2xδ + C1δ ≤

(2D2d0

+ C1

)δ ≤ 1

2d0 <12R′0(0) ,

and this gives 12R′0(0) ≤ R′0(x) + S′(x) ≤ 3

2R′0(0). Moreover, xδ ≤ |x| ≤ x0 implies

∣∣R0(x) + S(x)∣∣ ≥ ∣∣∣∣ˆ x

0R′0(s) ds

∣∣∣∣− ‖S′‖∞ |x| ≥ (d0 − C1δ) |x| > 12d0 ·

2

d0δ = δ ,

whereas for |x| > x0 we find ∣∣R0(x) + S(x)∣∣ ≥ r0 − C1δ

2 ≥ 12r0 ≥ δ .

Using

x− := max{x : R0(x) + S(x) ≤ −δ} , x+ := min{x : R0(x) + S(x) ≥ +δ} ,

we now easily verify that S is δ-admissible.Step 2: We next show that Xδ is invariant under T for δ < δ0. Since each S ∈ Xδ is δ-admissible,

Corollary 11 yields

‖AG(S)‖∞ ≤ CGδ , ‖A2G(S)‖∞ ≤ CG(1 + C2)δ2 ,

and ‖G(S)‖∞ ≤ 1 + CΨ holds by definition of G and Assumption 1. Lemma 9 now provides T (S)and η(S), as well as the estimates

‖T (S)‖∞ ≤ CMCG(1 + C2)δ2 = C0δ2 ,

‖T (S)′‖∞ ≤ CMCGδ = C1δ ,‖T (S)′′‖∞ ≤ CM(1 + CΨ) = C2 ,

and |η(S)| ≤ C0δ2. In particular, we have T (S) ∈ Xδ.

Theorem 14. For sufficiently small δ, the operator T has a unique fixed point in Xδ.

Proof. We equip Xδ with the norm ‖S‖# = ‖S‖∞+‖S′‖∞+δ‖S′′‖∞, which is, for fixed δ, equivalentto the standard norm. For given S1, S2 ∈ Xδ, we now employ the estimates from Lemma 9 andLemma 12. This gives

‖T (S2)− T (S1)‖# ≤ CM(‖A2G(S2)−A2G(S1)‖∞ + ‖AG(S2)−AG(S1)‖∞ + δ‖G(S2)− G(S1)‖∞

)≤ CMCLδ‖S2 − S1‖# ,

and we conclude that T is contractive with respect to ‖·‖# provided that δ < 1/(CMCL). The claimis now a direct consequence of the Banach Fixed Point Theorem.

We now show that different anchors provide different waves.

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Lemma 15. Let (R0, µ0) and (R0, µ0) be two different travelling waves for δ = 0 and the same valueof c, and (S, η) and (S, η) be the corresponding correctors provided by Theorem 14. Then R0 + S0

and R0 + S are different.

Proof. By construction, we have

S, S ∈ U1 := span {Y1, Y2, 1} ⊕ L2(R) ,

while Proposition 2 guarantees that

R0 − R0 ∈ U2 := span {sin (kc·), 1− cos (kc·)} .

The claim now follows since U1 ∩ U2 = {0}.

4 Change in the kinetic relation

In this final section we show that the kinetic relation does not change to order O(δ). To this endwe denote by Rδ a travelling wave solution to (2) as provided by Theorem 3. The correspondingconfigurational force, cf. [HSZ12], is then defined by Υδ := Υe,δ −Υf,δ with

Υe,δ := Φδ(rδ,+)− Φδ(rδ,−) , Υf,δ :=Φ′δ(rδ,+) + Φ′δ(rδ,−)

2

(rδ,+ − rδ,−

),

where the macroscopic strains rδ,± on both sides of the interface can be computed from Rδ via

rδ,± = limL→∞

1

L

ˆ +L

0Rδ(±x) dx .

Lemma 16. Let Rδ be a travelling wave from Theorem 3, and R0 the corresponding wave for δ = 0.Then we have Υδ = Υ0 +O(δ2).

Proof. By construction, we know that the only asymptotic contributions to the profile Rδ are due toR0 − Iδ plus a small asymptotic corrector of order O(δ2) from span{1, Y1, Y2}. This implies

rδ,± = r0,± − Iδ +O(δ2) .

As r0,± and rδ,± are both larger than δ we know that

Ψ′δ(rδ,±) = ∓1 = Ψ′0(rδ,±).

Thus, we conclude

Φ′δ(rδ,±) = rδ,± ∓ 1 = Φ′0(r0,±)− Iδ +O(δ2) ,

and hence

Υf,δ = Υf,0 − Iδ(r0,+ − r0,−) +O(δ2) .

Moreover, we calculate

Υe,δ =

ˆ rδ,+

rδ,−

Φ′δ(r) dr =

ˆ rδ,+

rδ,−

(r −Ψ′δ(r)−Ψ′0(r) + Ψ′0(r)

)dr

=

ˆ rδ,+

rδ,−

Φ′0(r) dr −ˆ rδ,+

rδ,−

(Ψ′δ(r)−Ψ′0(r)

)dr

= Φ0(rδ,+)− Φ0(rδ,−)− 2Iδ = 12(rδ,+ − 1)2 − 1

2(rδ,− + 1)2 − 2Iδ

= 12(r0,+ − Iδ − 1)2 − 1

2(r0,− − Iδ + 1)2 − 2Iδ +O(δ2)

= 12(r0,+ − 1)2 − 1

2(r0,− + 1)2 − Iδ(r0,+ − 1− r0,− − 1)− 2Iδ +O(δ2)

= Υe,0 − Iδ(r0,+ − r0,−) +O(δ2) .

Subtracting both results gives Υδ = Υ0 +O(δ2), the desired result.

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Acknowledgement

The authors gratefully acknowledge financial support by the EPSRC (EP/H05023X/1).

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